Feynman spectral action of the wave operator on asymptotically de Sitter spaces

TL;DR

This paper develops a Feynman spectral action for the wave operator on asymptotically de Sitter spaces, providing uniform microlocal estimates and enabling spectral analysis in a Lorentzian setting.

Contribution

It introduces a Feynman operator on conformal extensions of asymptotically de Sitter spaces and proves uniform microlocal estimates for it, advancing spectral analysis in Lorentzian geometry.

Findings

Constructed a Feynman operator on conformal extensions.

Proved uniform microlocal estimates for the Feynman operator.

Enabled study of Lorentzian spectral zeta functions and spectral actions.

Abstract

In this paper, we investigate the wave operator $\square_g$ on non-trapping (at all energies) even asymptotically de Sitter spaces. We construct a Feynman operator on the conformal extension of asymptotically de Sitter spaces and give a proof of uniform microlocal estimates for the Feynman operator in this setting. This enables the study of the Lorentzian "spectral" zeta functions in asymptotically de Sitter and the construction of a "spectral" action of the Feynman propagator.